Question

A metal ring carries a current that causes a magnetic field ${\mathbf{B}}_{\mathbf{0}}$at the center of the ring and a field B at point P a distance xfrom the center along the axis of the ring. If the radius of the ring is doubled, find the magnetic field at the center. Will the field at point P change by the same factor? Why?

Short answer

No, the field at point P does not change by the same factor. And the magnetic field at the center when radius of the loop doubled${\mathrm{B}}_0=\frac{1}{2}{\mathrm{B}}_0$

Step-by-step solution

Definition of the current

The term current may be defined as the ratio of charge and time.

Determine the magnetic field and effect on field by the same factor.

At the center of circular current carrying wire with radius $r$the magnetic field IS ${\mathrm{B}}_0=\frac{{\mathrm{\mu }}_0\mathrm{l}}{2\mathrm{r}}$and magnetic field for the distance from the center of the loop, along the axis of ring is

$\mathrm{B}=\frac{{\mathrm{\mu }}_0}{2}\left(\frac{{\mathrm{Ir}}^2}{{\left({\mathrm{r}}^2+{\mathrm{x}}^2\right)}^{3/2}}\right)$

And at a distance x from the center of the loop, the field is $\mathrm{B}\text{'}=\frac{{\mathrm{\mu }}_0}{2}\left[\frac{\mathrm{i}{\left(2\mathrm{r}\right)}^2}{{\left[{\left(2\mathrm{r}\right)}^2+{\mathrm{x}}^2\right]}^{3/2}}\right]\frac{{}_{1}1}{2}\mathrm{B}$

Hence,no, the field at point P does not change by the same factor. And the magnetic field at the center when radius of the loop doubled $\mathrm{B}{\text{'}}_0=\frac{1}{2}{\mathrm{B}}_0$.